Wei Zhang, Jie Fu, Chengchun Hao and Siqi Yang
Global well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximationThis paper focuses on the global well-posedness of the Oberbeck-Boussinesq approximation for the unsteady motion of a drop in another bounded fluid separated by a closed interface with surface tension. We assume that the initial states of the drop are close to a ball $ B_{R}$ with the same volume as the drop, and that the boundary of the drop is a small perturbation of the boundary of $ B_{R}$. To begin, we introduce the Hanzawa transformation with an added barycenter point to obtain the linearized Oberbeck-Boussinesq approximation in a fixed domain. Then, we establish time-weighted estimates of solutions for the shifted equation using maximal $ L^p$-$ L^q$ regularities for the two-phase fluid motion of the linearized system, as obtained by Hao and Zhang in 2022. Using the time decay estimates of the semigroup, we obtain decay time-weighted estimates of solutions for the linearized problem. Additionally, we prove that these estimates are less than the sum of the initial value and its own square and cube by estimating the corresponding non-linear terms. Finally, the existence and uniqueness of solutions in the finite time interval $(0,T)$ was proven by Hao and Zhang in 2023. After that, we demonstrate that the solutions can be extended beyond $ T$ by analyzing the properties of the roots of algebraic equations.
@article{ZFHY24a,
author ={Zhang, Wei and Fu, Jie and Hao, Chengchun and Yang, Siqi},
journal={Journal of Mathematical Physics},
title ={Global well-posedness for two-phase fluid motion in the Oberbeck-Boussinesq approximation},
year ={2024},
volume ={65},
number ={8},
pages ={081509 (22pp)},
doi={10.1063/5.0220764},
}